Quantum Ultra-Walks: Walks on a Line with Hierarchical Spatial Heterogeneity

We discuss the model of a one-dimensional, discrete-time walk on a line with spatial heterogeneity in the form of a variable set of ultrametric barriers. Inspired by the homogeneous quantum walk on a line, we develop a formalism by which the classical ultrametric random walk as well as the quantum walk can be treated in parallel by using a "coined" walk with internal degrees of freedom. For the random walk, this amounts to a $2^{{\rm nd}}$-order Markov process with a \emph{stochastic} coin, better known as an (anti-)persistent walk. When this coin varies spatially in the hierarchical manner of "ultradiffusion," it reproduces the well-known results of that model. The exact analysis employed for obtaining the walk dimension $d_{w}$, based on the real-space renormalization group (RG), proceeds virtually identical for the corresponding quantum walk with a $unitary$ coin. However, while the classical walk remains robustly diffusive ($d_{w}=\frac{1}{2}$) for a wide range of barrier heights, unitarity provides for a quantum walk dimension $d_{w}$ that varies continuously, for even the smallest amount of heterogeneity, from ballistic spreading ($d_{w}=1$) in the homogeneous limit to confinement ($d_{w}=\infty$) for diverging barriers. Yet for any $d_{w}<\infty$ the quantum ultra-walk never appears to localize.